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# How do you simplify ${(x - y)^3}?$

Last updated date: 03rd Mar 2024
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Hint: To simplify the given expression we will multiply the term $(x - y)$ three times in an algebraic mathematical process. On doing calculations we get the required answer.

Formula used: Here we will use the following indices formula:
${a^{m + n}} = {a^m}{a^n}$

Complete Step by Step Solution:
We have to simplify ${(x - y)^3}$ in a formation of addition or subtraction with variables.
Now, the term $(x - y)$ is up to the power of $3$.
So, we can rewrite the expression as: ${(x - y)^{1 + 1 + 1}}$.
Now, we can use the above indices formula.
As the expression given is ${(x - y)^3}$, it is clearly visible that we can multiply $(x - y)$ three times to achieve the final expression.
So, we can rewrite the expression as following:
$\Rightarrow {(x - y)^3} = (x - y) \times (x - y) \times (x - y).$
So, we will multiply first two $(x - y)$’s then the result of it will multiply by another $(x - y)$ to get the final answer.
Therefore, the multiplication of $(x - y) \times (x - y)$ is as following:
$\Rightarrow (x - y) \times (x - y) = (x - y) \times x - (x - y) \times y$.
So, after the multiplication of these terms, we get the following expression:
$\Rightarrow ({x^2} - xy) - (xy - {y^2})$.
Now, if we take the brackets off from the above expression, we get the following expression:
$\Rightarrow {x^2} - xy - xy + {y^2}$.
Now, if we arrange the middle terms and other terms accordingly, we get the following expression:
$\Rightarrow {x^2} - 2xy + {y^2}$.
So, we can state that :
$\Rightarrow {(x - y)^2} = (x - y) \times (x - y)$
Therefore, ${(x - y)^2} = {x^2} - 2xy + {y^2}$.
Now, to get the final value of the given expression${(x - y)^3}$, we have to multiply ${(x - y)^2}$ by $(x - y)$
Therefore, we can write the following expression:
$\Rightarrow {(x - y)^3} = {(x - y)^2} \times (x - y)$.
So, we can rewrite it as following:
$\Rightarrow {(x - y)^3} = (x - y) \times ({x^2} - 2xy + {y^2})$.
So, if we multiply the above terms and split up, we get:
$\Rightarrow ({x^2} - 2xy + {y^2}) \times x - ({x^2} - 2xy + {y^2}) \times y$.
Now, if we multiply it further, we get:
$\Rightarrow ({x^2} \times x - 2xy \times x + {y^2} \times x) - ({x^2} \times y - 2xy \times y + {y^2} \times y)$.
Now, by doing the further simplification, we get:
$\Rightarrow ({x^3} - 2{x^2}y + x{y^2}) - ({x^2}y - 2x{y^2} + {y^3})$.
Now, if we take the brackets off, we get:
$\Rightarrow {x^3} - 2{x^2}y + x{y^2} - {x^2}y + 2x{y^2} - {y^3}$.
Now, add or subtract the terms having same degrees, we get:
$\Rightarrow {x^3} - 3{x^2}y + 3x{y^2} - {y^3}$.
We can rearrange the above statement in following way:
$\Rightarrow {x^3} - {y^3} - 3xy(x - y)$.

$\therefore$The answer of the question is ${x^3} - {y^3} - 3xy(x - y)$.

Note: Points to remember:
If the power of any expression or term is $n$, where $n$ is any natural number, then the expression shall multiply $n$ numbers of times.
Always expand each term in the bracket by all the other terms in the other brackets, but never multiply two or more terms in the same bracket.